My system is from time-integration of the semi-discretized Stokes equation. The time update of the variables $(v,p)$ is defined via the solution of
$$ \begin{bmatrix} A & -\tau B^T \\ B & 0 \end{bmatrix} \begin{bmatrix} v \\p \end{bmatrix} = f \quad\quad (*) $$ or, equivalently, $$ \begin{bmatrix} A & B^T \\ B & 0 \end{bmatrix} \begin{bmatrix} v \\ \tilde p \end{bmatrix} = f, \quad\quad (**) $$ where $\tilde p := -\tau p$.
Note that, since $A$ is assumed symmetric, the coefficient matrix in $(**)$ is symmetric.
Say, I start with a zero initial value and I use GMRes to solve $(*)$ and MinRes to solve $(**)$ iteratively.
Question: Do the iterates -- after rescaling $p:=-\frac{1}{\tau}\tilde p$ -- of both iteration schemes coincide?
If not, can I say something about the approximation quality of $p$ in the MinRes iteration for $(**)$ taking into account that $\tau \ll 1$.
Short answer: No, the iterates do not coincide, see the numerical experiment with KryPy. Note that GMRES can be used in both cases since MINRES is (mathematically) equivalent to GMRES.
Long answer: applying GMRES to $(**)$ and correcting afterwards is equivalent to applying GMRES to $(*)$ with the right preconditioner $$ M_r= \begin{bmatrix} I & 0\\ 0& -\tau I \end{bmatrix}. $$ In general, the Krylov subspace $K_n(C,f)$ for $(*)$ does not equal $K_n(CM_r,f)$ for $(**)$ (where $C$ is the block matrix in $(*)$). Thus, the iterates will not coincide in general. Furthermore, I am not aware of any possibility to relate the iterates in a meaningful way.